I need a birational transformation of singular quartics into weierstrass form. the quartics are of genus one and have the following form: $$(x^2 + c) (y^2 + i) + k = 0$$ where $c,i,k$ are given rational numbers, and $k$ is of big height. The only known rational solutions: the singular points in homogenous coordinates: $(x,y,z)=(0,1,0),(1,0,0)$.
Methods described in the literature (e.g. Lawrence Washington, Elliptic curves, p.37) require a non-singular rational solution. but after this text a singular quartic can be transformed birationally into a nonsingular curve, where the singular solutions are mapped into two nonsingular solutions. Unfortunately, I have not found in this work and in google, how to do this transformation. I ask for information, how to transform singular quartics as above into Weierstrass form or how to transform into a nonsingular curve.